On the Isomorphism Problem of P-endomorphisms Table of Contents

On the Isomorphism Problem of p-Endomorphisms Peter Jong, Ph.D. Department of Mathematics, University of Toronto, 2003 Let X = (X,B, μ, T ) be a measure-preserving system on a Lebesgue probability space. Given a fixed probability vector p = (p1, . . . , ps), we say that X = (X,B, μ, T ) is a pendomorphism if T is s-to-1 a.e. and the conditional probabilities of the preimages are precisely the components of p. Two measure-preserving systems X = (X,B, μ, T ) and Y = (Y, C, ν, S) are isomorphic if there exist a measure-preserving bijective map φ : X → Y such that φT = Sφ a.e. This thesis considers the isomorphism problem of p-endomorphisms, generalizing the work of Hoffman and Rudolph [H,R] which treats the case when p is a uniform probability vector, i.e. p = (1/p, . . . , 1/p). In particular, we generalize the tvwB criterion introduced in Hoffman and Rudolph to prove two results. The first result is Theorem 2.4.1, which generalizes the main theorem in [H,R] to pendomorphisms. We paraphrase this as follows: Theorem 2.4.1. Let X = (X,B, μ, T ) be a p-endomorphism. Then X = (X,B, μ, T ) is one-sided Bernoulli if and only if X = (X,B, μ, T ) is tvwB. We give two proofs of this result. The first follows Ornstein’s classical proof of his famous theorem that two shifts of equal entropy are isomorphic, and a second proof which follows the joinings proof as given in [H,R]. As a corollary of the joinings proof, we show that there are uncountably many automorphisms of the one-sided Bernoulli shift B(p) unless the components of p are pairwise distinct. We also give examples of tvwB p-endomorphisms such as mixing one-sided Markov shifts and a generalization of the [T, Id] transformation. The second main result is Theorem 5.1.1, which in view of Theorem 2.4.1, reduces to the statement that for any two tvwB finite group extensions of one-sided Bernoulli shifts, there is an isomorphism between them in a stronger sense than that asserted in Theorem 2.4.1. Specifically, we have the following theorem in Chapter 5 which we paraphrase as follows: Theorem 5.1.1. Let G be a finite group. For any two tvwB G-extensions of the one-sided shift B(p), there is an isomorphism which preserves the Bernoulli factor algebra and maps fibres over points in the factor to other such fibres by group rotations.